A340335 E.g.f.: Sum_{n>=0} x^n * exp(2^n*x).

1, 2, 7, 43, 441, 7241, 185233, 7252337, 429318529, 38079107713, 5026601726721, 982190576713985, 282875400939199489, 119655020273311783937, 74116220750762851756033, 67059149595727519091988481, 88432643551731676843100438529

Offset

0,2

Comments

The e.g.f. A(x) of this sequence is motivated by the following identity:

Sum_{n>=0} p^n * exp(q*r^n) = Sum_{n>=0} (q^n/n!) / (1 - p*r^n) ;

here, p = x, q = x, and r = 2.

Links

Table of n, a(n) for n=0..16.

Formula

E.g.f.: Sum_{n>=0} x^n * exp(2^n*x).

E.g.f.: Sum_{n>=0} (x^n/n!) / (1 - 2^n*x).

Example

E.g.f.: A(x) = 1 + 2*x + 7*x^2/2! + 43*x^3/3! + 441*x^4/4! + 7241*x^5/5! + 185233*x^6/6! + 7252337*x^7/7! + 429318529*x^8/8! + 38079107713*x^9/9! + ...
where
A(x) = exp(x) + x*exp(2*x) + x^2*exp(2^2*x) + x^3*exp(2^3*x) + x^4*exp(2^4*x) + x^5*exp(2^5*x) + ...
also
A(x) = 1/(1 - x) + x/(1 - 2*x) + (x^2/2!)/(1 - 2^2*x) + (x^3/3!)/(1 - 2^3*x) + (x^4/4!)/(1 - 2^4*x) + (x^5/5!)/(1 - 2^5*x) + ...

Prog

(PARI) {a(n) = my(A = sum(m=0, n, x^m * exp(2^m*x +x*O(x^n)) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = my(A = sum(m=0, n, x^m/m! * 1/(1 - 2^m*x +x*O(x^n)) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A340450.

Sequence in context: A014501 A197910 A228230 * A011835 A198946 A212270

Adjacent sequences: A340332 A340333 A340334 * A340336 A340337 A340338

Keyword

nonn

Author

Paul D. Hanna, Jan 11 2021

Status

approved